Semantics of Time Varying Attributes and Their
Use for Temp oral Database Design
Christian S Jensen Richard T Sno dgrass
1
Department of Mathematics and Computer Science Aalb org University Fredrik
Ba jers Vej E DK Aalb org DENMARK csj iesd auc dk
2
Department of Computer Science University of Arizona Tucson AZ USA
rts cs arizona edu
Abstract Based on a systematic study of the semantics of temp oral
attributes of entities this pap er provides new guidelines for the design
The notions of observation and up date of temp oral relational databases
patterns of an attribute capture when the attribute changes value and
when the changes are recorded in the database A lifespan describ es
when an attribute has a value And derivation functions describ e how
the values of an attribute for all times within its lifespan are computed
from stored values The implications for temp oral database design of the
semantics that may b e captured using these concepts are formulated as
schema decomp osition rules
Intro duction
Designing appropriate database schemas is crucial to the e ective use of rela
tional database technology and an extensive theory has b een develop ed that
sp eci es what is a go o d database schema and how to go ab out designing such
a schema The relation structures provided by temp oral data mo dels e g the
recent TSQL mo del provide built in supp ort for representing the temp o
ral asp ects of data With such new relation structures the existing theory for
e e ective use of tem relational database design no longer applies Thus to mak
p oral database technology a new theory for temp oral database design must b e
develop ed
We have previously extended and generalized conventional normalization
But the resulting concepts are still lim concepts to temp oral databases
ited in scop e and do not fully account for the time varying nature of data
Thus additional concepts are needed in order to fully capture and exploit the
time varying nature of data during database design This pap er prop oses con
cepts that capture the time related semantics of attributes and uses these as a
foundation for developing guidelines for the design of temp oral databases The
prop erties of time varying attributes are captured by describing their lifespans
their time patterns and their derivation functions Design rules subsequen tly al
low the database designer to use the prop erties for view logical and physical
schema design
The pap er is structured as follows Section rst reviews the temp oral data
mo del used in the pap er It then argues that the prop erties of attributes are rela
they describ e and then intro duces surrogates for representing tive to the ob jects
real world ob jects in the mo del The following subsections address in turn dif
ferent asp ects of time varying attributes namely lifespans time patterns and
derivation functions Section is devoted to the implications of the attribute se
mantics for logical schema physical schema and view design The nal section
summarizes and p oints to opp ortunities for further research
Capturing the Semantics of Time Varying Attributes
This section provides concepts that allow the database designer to capture more
precisely and concisely than hitherto the time varying nature of attributes in
temp oral relations The temp oral data mo del employed in the pap er is rst
describ ed Then a suite of concepts for capturing the temp oral semantics of
attributes are intro duced
A Conceptual Data Mo del
We describ e brie y the relation structures of the Bitemp oral Conceptual Data
Mo del BCDM see for a more complete description that is the data mo del
of TSQL and which is used in this pap er
We adopt a linear discrete b ounded mo del of time with a time line com
p osed of chronons The schema of a bitemp oral conceptual relation R consists
of an arbitrary numb er e g n of explicit attributes and an implicit timestamp
bitemp oral attribute T de ned on the domain of sets of bitemp oral chronons A
b t v t
chronon c c c is an ordered pair of a transaction time chronon c and a
v b
valid time chronon c A tuple x a a j t in a relation instance r a
n
of schema R thus consists of n attribute values asso ciated with a bitemp oral
timestamp value An arbitrary subset of the domain of valid times is asso ciated
with each tuple meaning that the information recorded by the tuple is true in
the modeled reality during each valid time chronon in the subset Each individ
ual valid time chronon of a single tuple has asso ciated a subset of the domain
of transaction times meaning that the information valid during the particular
ent in the relation during each of the transaction time chronons chronon is curr
in the subset Any subset of transaction times less than the current time may
that while the de nition of a bitemp oral b e asso ciated with a valid time Notice
chronon is symmetric this explanation is asymmetric re ecting the di erent
semantics of transaction and valid time
We have thus seen that a tuple has asso ciated a set of so called bitemp oral
chronons in the two dimensional space spanned by transaction time and valid
time Such a set is termed a bitemporal element We assume that a do
main of surrogate values is available for representing real world ob jects in the database
Example Consider the relation instance empDep shown next
EName Dept T
Bob Ship f
g
Bob Load f g
The relation shows the employment information for an employee Bob and two
departments Ship and Load contained in two tuples In the timestamps we
assume that the chronons corresp ond to days and that the p erio d of interest is
Throughout we use integers some given month in a given year e g July
as timestamp comp onents The reader may informally think of these integers as
dates e g the integer in a timestamp represents the date July The
current time is assumed to b e
V alid time relations and transaction time relations are sp ecial cases of bitem
p oral relations that supp ort only valid time and transaction time resp ectively
or clarity we use the term snapshot relation for a conventional relation which F
supp orts neither valid time nor transaction time
This completes the description of the ob jects in the bitemp oral conceptual
data mo del relations of tuples timestamp ed with temp oral elements An asso
ciated algebra and user level query language are de ned elsewhere
Using Surrogates
An attribute is seen in the context of a particular real world entity Thus when
we talk ab out a prop erty e g the frequency of change of an attribute that
prop erty is only meaningful when the attribute is asso ciated with a particular
entity As an example the frequency of change of a salary attribute with re
sp ect to a sp eci c employee in a company may reasonably b e exp ected to b e
relatively regular and there will only b e at most one salary for the employee at
each p oint in time In contrast if the salary is with resp ect to a department a
signi cantly di erent pattern of change may b e exp ected There will generally
b e many salaries asso ciated with a department at a single p oint in time Hence
it is essential to identify the reference ob ject when discussing the semantics of
an attribute
We employ surrogates for representing real world entities in the database
In this regard we follow the approach adopted in e g the TEER mo del by
Elmasri Surrogates do not vary over time in the sense that two entities iden
ti ed by identical surrogates are the same entity and two entities identi ed by
di erent surrogates are di erent entities We assume the presence of surrogate
attributes throughout logical design At the conclusion of logical design surro
gate attributes may b e either retained replaced by regular key attributes or
eliminated
Lifespans of Individual Time Varying Attributes
In database design one is interested in the interactions among the attributes of
the relation schemas that make up the database
Here we provide a basis for relating the lifespans of attributes Intuitively the
lifespan of an attribute for a sp eci c ob ject is all the times when the ob ject has
inapplicable null for the attribute Note that lifespans a value distinct from
i
concern valid time i e are ab out the times when there exist some valid values
To more precisely de ne lifespans we rst de ne an algebraic selection op er
ator on a temp oral relation De ne a relation schema R A A jT and
n
b e a predicate de ned on the A The let r b e an instance of this schema Let P
i
B B
selection P on r r is de ned by r fz j z r P z A A g
n
P P
B
It follows that r simply p erforms the familiar snapshot selection with the
P
addition that each selected tuple carries along its timestamp T Next we de ne
an auxiliary function vte that takes as argument a valid time relation r and
v v
returns the valid time element de ned by vte r fc j s s r c s T g
The result valid time element is thus the union of all valid timestamps of the
tuples in an argument valid time relation
Let a relation schema R S A A j T b e given where S De nition
n
is surrogate valued and let r b e an instance of R The lifespan for an attribute
A i n with resp ect to a value s of S in r is denoted ls r A s and is
i i
B
de ned by ls r A s vte r
i
s^A6 ? S
i
Lifespans are imp ortant b ecause attributes are guaranteed to not have any
inapplicable null value during their lifespans Assume that we are given a relation
EName Dept that records the names and departments schema empDep EmpS
of employees represented by the surrogate attribute EmpS If employees always
have a name when they have a department and vice versa this means that
inapplicable nulls are not present in instances of the schema With lifespans this
prop erty may b e stated by saying that for all meaningful instances of EmpSal
and for all EmpS surrogates attributes EName and Dept have the same lifespans
The imp ortance of lifespans in temp oral databases has b een recognized in the
context of data mo dels in the past c f Our use of lifespans for database
design di ers from the use of lifespans in database instances In particular using
lifespans during database design do es not imply any need for storing lifespans
in the database
Time Patterns of Individual Time Varying Attributes
In order to capture how an attribute varies over time we intro duce the concept
of a time pattern Informally a time pattern is simply a sequence of times
is a partial function from the natural numb ers De nition The time pattern T
N to a domain D of times T N D If T i is de ned so is T j for
T T
i We term T i the i th time p oint all j
In the context of databases two distinct typ es of time patterns are of partic
ular interest namely observation patterns and up date patterns The observation
s
pattern O for an attribute A relative to a particular surrogate s is the times
A
when the attribute is given a particular value p erhaps as a result of an observa
tion e g if the attribute is sampled a prediction or an estimation We adopt
s
the convention that O is the time when it was rst meaningful for attribute
A
A to have a value for the surrogate s Observation patterns concern valid time
The observation pattern may b e exp ected to b e closely related to but distinct
from the actual p ossibly unknown pattern of change of the attribute in the
s
is the times when the value of the at mo deled reality The update pattern U
A
tribute is up dated in the database Thus up date patterns concern transaction
time
Note that an attribute may not actually change value at a time p oint b ecause
it may b e the case that the existing and new values are the same The times
when changes take place and the resulting values are orthogonal asp ects In the
latter half of Section we will return to this distinction
The Values of Individual Time Varying Attributes
We pro ceed by considering how attributes may enco de information ab out the
ob jects they describ e As the enco ding of the transaction time of attributes is
typically built into the data mo del we consider only valid time relations
A relation may record directly when a particular attribute value is valid
Alternatively what value is true at a certain p oint in time may b e computed
from the recorded values In either case the relation is considered a valid time
relation An example clari es the distinction b etween the two cases
Example Consider the two relations shown b elow The rst empSal records
names and salaries of employees and the second expTemp records names and
temp erature measurements for exp eriments Attributes EmpS and ExpS record
surrogates representing employees and exp eriments resp ectively
EmpS EName Sal T ExpS Exp Temp T
e Bob k f g x Exp f g
e Bob k f g x Exp f g
e Bob k f g x Exp f g
e Bob k f g x Exp f g
e Sam k f g x Exp f g
e Sam k f g x Exp f g
empSal expTemp
Relation empSal records Bob s and Sam s salaries at all the times they have
salaries This is clearly consistent with what a valid time relation is At rst sight
relation expTemp is more problematic It do es not app ear to record temp eratures
for all the times when there exists a temp erature for exp eriment x Sp eci cally
we may envision that the temp erature of x is sampled regularly and that we
may later want to compute x temp erature values for times with no explicitly
recorded value
Traditionally empSal has b een considered a state relation and expTemp has
most data mo del prop osals with notable b een considered an event relation
exceptions e g have considered only the rst typ e of relation
However note that the relations are similar in the sense that they b oth record
when information is true Due to this observation we make no fundamental
distinction b etween the two typ es of relations but instead treat them quite
similarly
The di erence b etween relations such as empSal and expTemp in the example
ab ove is solely in what additional or even di erent information is implied by
each of the relations At the one extreme relation empSal do es not imply any
additional information at all No salary is recorded for Bob from time to time
and the existing tuples do not imply any salary for Bob in that time interval
The other sample relation is di erent For example while no temp erature for
Exp at time is recorded clearly such a temp erature exists Further we may
even have a go o d idea what the temp erature may b e i e close to
Thus the di erence is that di erent derivation functions apply to the salary
for and temp erature attributes of the two relations A derivation function f
A
a sp eci c attribute A of a relation schema R takes as arguments a valid time
v
chronon c and a relation instance r and returns a value in the domain of at
tribute A For the salary attribute a discrete derivation function applies and for
the temp erature a nearest neighb or derivation function may satisfy some users
while other users may need a more sophisticated function
De nition A derivation function f is a partial function from the domains of
and relation instances r with schema R to a value domain D valid times D
T V
in the universal set of domains D i e f D r R D
D VT
The imp ortance of derivation functions in data mo dels has previously b een
argued convincingly by e g Klopprogge Cli ord and Segev They
should thus also b e part of a design metho dology
Summary of Attribute Semantics
In summary the database designer is exp ected to initially identify and mo del
entity typ es using surrogates Then the notions of lifespans time patterns and
derivation functions are used for capturing the semantics of attributes
Elsewhere we have generalized conventional functional dep endencies to tem
p oral databases Essentially a temporal dependency holds on a temp oral re
lation if the corresp onding snapshot dep endency holds on each snapshot relation
contained in the temp oral relation With this generalization conventional rela
tional dep endency theory applies wholesale to temp oral databases For example
temporal keys may b e de ned Such keys are generally time varying As a basis
for de ning time invariant attributes and keys we have also de ned so called
strong temporal functional dependencies and strong temporal key While not
discussed here the designer is also exp ected to identify temp oral and strong
temp oral functional dep endencies
Temp oral Relational Database Design Guidelines
In this section we discuss how the prop erties of schemas with time varying
attributes as captured in the previous section are used during database design
Emphasis is on the implications of the prop erties for design of the logical schema
but implications for view design and physical design are touched up on as well
Logical Design Guidelines
Two imp ortant goals of logical database design are to design a database schema
that do es not require the use of inapplicable nulls and avoids representation of
the same information We de ne two prop erties that illuminate these asp ects of
relation schemas and guide the database designer
Database designers are faced with a numb er of design criteria which are
sometimes con icting making database design a challenging task So while we
discuss certain design criteria in isolation it is understo o d that there may b e
other criteria that should b e taken into consideration during database design
such as minimizing the impact of joins required on relations that have b een
decomp osed
Lifespan Decomp osition Rule One imp ortant design criterion in conven
tional relational design is to eliminate the need for inapplicable nulls in tuples of
database instances In the context of temp oral databases we use the notion of
lifespans to capture when attributes are de ned for the ob jects they are intro
duced in order to describ e Brie y the lifespan for an attribute with resp ect
to a particular surrogate representing the ob ject describ ed by the attribute is
all the times when a meaningful attribute value known or unknown exists for
the ob ject
Inapplicable nulls may o ccur in a relation schema when two attributes have
To identify this typ e of situa di erent lifespans for the same ob ject surrogate
tion we intro duce the notion of lifespan equal attributes Examples follow the
the de nition
De nition Let a relation schema R S A A j T b e given where S is
n
surrogate valued Two attributes A and A in R are termed lifespan equal with
i j
LS
resp ect to surrogate S denoted A A if for all meaningful instances r of R
i S j
s dom S ls r A s ls r A s
i j
To exemplify this de nition consider a relation schema Emp with attributes EmpS
and MgrSince The schema is used by a employee surrogates Dept Salary
company where each employee is always assigned to some department and has a
salary In addition the relation records when an employee in a department rst
b ecame a manager in that department
LS
For this schema we have Dept Salary b ecause an employee has a salary
EmpS
it might b e unknown or zero exactly when asso ciated with a department Thus
no instances of Emp will have tuples with an inapplicable null value for one of
LS
Dept and Salary and not for the other Next it is not the case that Dept
EmpS
LS
MgrSince and by inference not the case that Salary MgrSince This is
EmpS
so b ecause employees often are asso ciated with a department where they have
never b een a manager Thus instances of Emp may contain inapplicable nulls
Sp eci cally the nulls are asso ciated with attribute MgrSince as the lifespan of
this attribute is shorter than that of Dept and Salary
Next observe that Dept and Salary b eing lifespan equal with resp ect to EmpS
do es not mean that all employees have the same lifespan for their department
or salary attribute Employees may have b een hired at di erent times and the
lifespans are thus generally di erent for di erent employees Rather the equality
is b etween the department and the salary lifespan for individual employees
The following de nition then characterizes temp oral database schemas with
instances that do not contain inapplicable nulls
De nition A relation schema R S A A j T where S is surrogate
n
LS
R A B valued is lifespan homogeneous if A B
S
These concepts formally tie the connection b etween the notion of lifespans
of attributes with the o ccurrence of inapplicable nulls in instances With them
we are in a p osition to formulate the Lifespan Decomp osition Rule
De nition Lifesp an Decomposition Rule To avoid inapplicable nulls in tem
p oral database instances decomp ose temp oral relation schemas to ensure life
span homogeneity
It is appropriate to brie y consider the interaction of this rule with the the
existing temp oral normal forms that also prescrib e decomp osition of relation
schemas Sp eci cally while the decomp osition that o ccurs during normalization
do es as a side e ect aid in eliminating the need for inapplicable nulls a database
schema that ob eys the temp oral normal forms may still require inapplicable nulls
in its instances To exemplify consider again the Emp schema and think of the
temp oral dep endencies on temp oral relations as regular dep endencies on the
corresp onding snapshot tables Here EmpS is a temp oral key and there are no
other non trivial dep endencies Thus the schema is in temp oral BCNF It is
also the case that Emp has no non trivial temp oral multi valued dep endencies
and it is thus also in temp oral fourth normal form In spite of this we saw that
EmpS Dept there are inapplicable nulls The solution is to decomp ose Emp
into Emp EmpS Dept Salary and Emp EmpS Salary MgrSince
MgrSince Both resulting relations are lifespan homogeneous
Synchronous Decomp osition Rule The synchronous decomp osition rule is
based on the notion of observation pattern and its ob jective is to eliminate a
particular kind of redundancy We initially exemplify this typ e of redundancy
Then we de ne the notion of synchronous attributes which leads to a de nition
of synchronous schemas and an accompanying decomp osition rule that are aimed
at avoiding this redundancy Finally we view synchronism in a larger context by
relating it to existing concepts and discuss the decomp osition rule s p ositioning
with resp ect to logical versus physical design
Example Consider the relation instance empDepSal that follows next record
ing departments and salaries for employees The schema for the relation is in
temp oral BCNF with the surrogate valued attribute EmpS b eing the only min
imal key and no other non trivial dep endencies Yet it may b e observed that
the salary k and the departments A and B are rep eated once once and four
times in the instance resp ectively These rep etitions are due to attributes Dept
and Salary having di erent observation patterns Sp eci cally the instance is
consistent with the patterns shown to the right of the instance
EmpS Dept Salary T
e A k f g
e
O
Dept
e B k f g
e B k f g
e
O
Salary
e B k f g
e B k f g
e A k f g
In combination these observation patterns imply the redundancy that may b e
observed in the sample instance Thus capturing during database design which
attributes of the same relation schema have di erent observation patterns is a
means of identifying this typ e of redundancy
To capture precisely the synchronism of attributes de ne T j to b e the re
t
striction of time pattern T to the valid time element t that is to include only
those times also contained in t
De nition De ne relation schema R S A A j T where S is surro
n
S
and gate valued Two attributes A and A in R with observation patterns O
i j
A
i
S
S
O A if for all meaningful are synchronous with resp ect to S denoted A
S j i
A
j
instances r of R and for all surrogates s
S S
j O j O
A A
s j ls r A s \ls r A s r A \ls r A s i ls
i i j j
Thus attributes are synchronous if their lifespans are identical when restricted
to the intersection of their lifespans With this de nition we can characterize
relations that avoid the redundancy caused by a lack of synchronism and then
state the Synchronous Decomp osition Rule
De nition De ne relation schema R S A A j T where S is surro
n
S
gate valued Relation R is synchronous if A A R A A
i j i S j
Synchronous Decomposition Rule To avoid rep etition of attribute De nition
values in temp oral relations decomp ose relation schemas until they are syn
chronous
Alternative notions of synchronism have previously b een prop osed for data
base design by Navathe and Ahmed Lorentzos and Wijsen While
these notions are stated with varying degrees of clarity and precision and are
de ned in di erent data mo del contexts they all seem to capture the same ba
sic idea namely that of value b ased synchronism which is di erent from the
synchronism prop osed in this pap er
To explain the di erence consider the relation instance shown next
S A A T
j i
s a a f g
1 2
s a a f g
1 3
In value based synchronism value changes of attributes must o ccur synchronous
ly for attributes to b e synchronous Consequently the relation instance implies
that the attributes A and A in its schema are not value synchronous and
i j
value based decomp osition is thus prescrib ed Next it may b e that the at
tributes in the relation have identical observation patters and that it just acci
dentally happ ened that the new value of A when b oth attributes were observed
i
at time was the same as its old value This means that the relation is consis
tent with attributes A and A b eing observation pattern based synchronous
i j
and the synchronous decomp osition rule do es then not apply To conclude the
value based and pattern based synchronisms are quite unrelated
Further it is our contention that using the concept of value based synchro
nism during database design is problematic Sp eci cally it seems quite rare that
the database designer can guarantee that at al l times in the future when two
attributes in a tuple of relation are up dated one of them do es not get a new value
that is identical to its old value Thus it app ears that decomp osition based on
value based synchronism e ectively and unnecessarily leads to a binary data
mo del in which all relations have just two attributes a time invariant attribute
and a single time varying attribute This is in contrast to the pattern based
decomp osition prescrib ed in this pap er
This study is carried out in the context of TSQL It is our contention that
in this context the synchronous decomp osition rule is relevant only to physical
database design Surely the redundancy that may b e detected using the syn
chronism concept is imp ortant when storing temp oral relations At the same
time this typ e of redundancy is of little consequence for the querying of logical
level relations using the TSQL query language Indeed it will often
adversely a ect the ease of formulating queries if logical level relations are de
comp osed solely based on a lack of synchronism In conclusion the presence of
synchronous attributes in a relation may a ect p erformance p ositively or neg
atively but it do es not negatively a ect correctness or the ease of formulating
queries and it is thus a non issue at the logical level
The widespread presence of asynchronous attributes in relation schemas has
b een used for motivating various attribute value timestamp ed data mo dels where
in relations time is asso ciated with attribute values rather than with tuples b e
cause these mo dels avoids the redundancy see e g Since this redun
dancy is not a problem in TSQL which employs tuple timestamp ed relations
asynchronous attributes is not strictly an argument for attribute value times
tamp ed mo dels
Finally the need for synchronism at the logical level has previously b een
claimed to make normal forms and dep endency theory inapplicable e g
The argument is that few attributes are synchronous meaning that relation
schemas must b e maximally decomp osed which leaves other normalization con
cepts irrelevant This claim do es not apply to our data mo del
For completeness it should b e mentioned that while the synchronism con
cepts presented in this section have concerned valid time similar concepts that
concern transaction time and employ up date patterns rather than observation
patterns may also b e de ned
Implications for View Design
The only concept from Section not covered so far is derivation functions These
relate to view design as outlined next
For each time varying attribute we have captured a set of one or more deriva
tion functions that apply to it It is often the case that exactly one derivation
function applies to an attribute namely the discrete interp olation function
that is a kind of identity function However it may also b e the case that several
nontrivial derivation functions apply to a single attribute
The problem is then how to apply several derivation functions to the base
data We feel that there should b e a clear separation b etween recorded data and
data derived from the stored data via some function Maintaining this separation
makes it p ossible to later mo dify existing interp olation functions
The view mechanism provides an ideal solution that maintains this separa
tion Thus the database designer rst identi es which sets of derivation func
tions that should b e applied simultaneously to the attributes of a logical relation
instance and then subsequently de nes a view for each such set Although in
terp olation functions have previously b een studied we b elieve they have never
b efore b een asso ciated with the view mechanism
Summary and Research Directions
In order to exploit the full p otential of temp oral relational database technology
guidelines for the design of temp oral relational databases should b e provided
This pap er has presented concepts for capturing the prop erties of time
varying attributes in temp oral databases These concepts include surrogates
that represent the real world ob jects describ ed by the attributes lifespans of
attributes observation and up date patterns for time varying attributes and
derivation functions that compute new attribute values from stored ones It
was subsequently shown how surrogates and lifespans play an role during design
of the logical database schema In particular the notion of lifespans led to the
formulation of a lifespan decomp osition rule The notion of observation and
up date patterns led to a synchronous decomp osition rule it was argued that
this rule should ideally apply to physical database design Finally it was shown
how derivation functions are relevant for view design
We feel that several asp ects merit further study An integration of the vari
ous existing contributions to temp oral relational database design into a coherent
framework has yet to b e attempted Likewise a complete design metho dology in
cluding conceptual implementation data model indep endent design and logical
design for temp oral databases is warranted Finally a next step is to adopt the
concepts provided in this pap er in richer entity based or semantic or ob ject
based data mo dels
wledgements Ackno
This work was supp orted in part by NSF grant ISI In addition the rst
author was supp orted in part by the Danish Natural Science Research Council
and grants
References
J Cli ord and A Croker The Historical Relational Data Mo del HRDM and
Algera Based on Lifespans In Proceedings of ICDE pp February
J Cli ord and A U Tansel On an Algebra for Historical Relational Databases
Two Views In Proceedings of ACM SIGMOD pp May
R Elmasri G Wuu and V Kourama jian ATemp oral Mo del and Query Lan
guage for EER Databases In pp
S K Gadia A Homogeneous Relational Mo del and Query Languages for Temp o
ral Databases ACM TODS Decemb er
S K Gadia and J H Vaishnav A Query Language for a Homogeneous Temp oral
Database In Proceedings of ACM PODS pp March
Hayes and S Ja jo dia eds C S Jensen J Cli ord R Elmasri S K Gadia P
A Glossary of Temp oral Database Concepts SIGMOD Record March
C S Jensen R T Sno dgrass and M D So o Extending Normal Forms to Tem
p oral Relations Technical Rep ort TR Department of Computer Science
University of Arizona Tucson AZ July
C S Jensen and R T Sno dgrass Semantics of Time Varying Attributes and
Their Use for Temp oral Database Design Technical Rep ort R Depart
ment of Math and Computer Science Aalb org University Denmark May
C S Jensen M D So o and R T Sno dgrass Unifying Temp oral Mo dels via a
Conceptual Mo del Information Systems
M R Klopprogge and P C Lo ckemann Mo delling Information Preserving Data
bases Consequences of the Concept of Time In Proceedings of VLDB pp
N A Lorentzos Management of Intervals and Temp oral Data in the Relational
Mo del Technical Rep ort Agricultural University of Athens
S B Navathe and R Ahmed ATemp oral Relational Mo del and a Query Lan
guage Information Sciences
R T Sno dgrass editor The TSQL Temp oral Query Language Kluwer Academic
Publishers xxiv pages
M D So o C S Jensen and R T Sno dgrass An Algebra for TSQL In
chapter pp
R T Sno dgrass The Temp oral Query Language TQuel ACM TODS
June
A Segev and A Shoshani ATemp oral Data Mo del based on Time Sequences In
pp
A U Tansel J Cli ord S K Gadia A Segev and R T Sno dgrass eds Temp oral
Databases Theory Design and Implementation Benjamin Cummings
J Wijsen Extending Dep endency Theory for Temp oral Databases Ph D Thesis
Department Computerwetenschapp en Katholieke Universiteit Leuven
a
This article was pro cessed using the L T X macro package with LLNCS style E